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Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight
line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
. Linearity is closely related to '' proportionality''. Examples in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
include
rectilinear motion Linear motion, also called rectilinear motion, is one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion, with co ...
, the linear relationship of
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
and current in an electrical conductor (
Ohm's law Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equat ...
), and the relationship of
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
and
weight In science and engineering, the weight of an object is the force acting on the object due to gravity. Some standard textbooks define weight as a Euclidean vector, vector quantity, the gravitational force acting on the object. Others define weigh ...
. By contrast, more complicated relationships are ''
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
''. Generalized for functions in more than one
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
, linearity means the property of a function of being compatible with
addition Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
and
scaling Scaling may refer to: Science and technology Mathematics and physics * Scaling (geometry), a linear transformation that enlarges or diminishes objects * Scale invariance, a feature of objects or laws that do not change if scales of length, energ ...
, also known as the
superposition principle The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So tha ...
. The word linear comes from
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
''linearis'', "pertaining to or resembling a line".


In mathematics

In mathematics, a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
or linear function ''f''(''x'') is a function that satisfies the two properties: *
Additivity Additive may refer to: Mathematics * Additive function, a function in number theory * Additive map, a function that preserves the addition operation * Additive set-functionn see Sigma additivity * Additive category, a preadditive category with fi ...
: . *
Homogeneity Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
of degree 1: for all α. These properties are known as the superposition principle. In this definition, ''x'' is not necessarily a
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
, but can in general be an element of any
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
. A more special definition of linear function, not coinciding with the definition of linear map, is used in elementary mathematics (see below). Additivity alone implies homogeneity for rational α, since f(x+x)=f(x)+f(x) implies f(nx)=n f(x) for any
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
''n'' by
mathematical induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ...  all hold. Informal metaphors help ...
, and then n f(x) = f(nx)=f(m\tfracx)= m f(\tfracx) implies f(\tfracx) = \tfrac f(x). The
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
of the rational numbers in the reals implies that any additive
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
is homogeneous for any real number α, and is therefore linear. The concept of linearity can be extended to linear operators. Important examples of linear operators include the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
considered as a
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
, and other operators constructed from it, such as del and the
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
. When a
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
can be expressed in linear form, it can generally be solved by breaking the equation up into smaller pieces, solving each of those pieces, and summing the solutions.
Linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
is the branch of mathematics concerned with the study of vectors,
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s (also called 'linear spaces'), linear transformations (also called 'linear maps'), and systems of linear equations. For a description of linear and nonlinear equations, see ''
linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficien ...
''.


Linear polynomials

In a different usage to the above definition, a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
of degree 1 is said to be linear, because the
graph of a function In mathematics, the graph of a function f is the set of ordered pairs (x, y), where f(x) = y. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in two-dimensional space and thus form a subset ...
of that form is a straight line. Over the reals, a
linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficien ...
is one of the forms: :f(x) = m x + b\ where ''m'' is often called the
slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
or
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
; ''b'' the
y-intercept In analytic geometry, using the common convention that the horizontal axis represents a variable ''x'' and the vertical axis represents a variable ''y'', a ''y''-intercept or vertical intercept is a point where the graph of a function or relatio ...
, which gives the point of intersection between the graph of the function and the ''y''-axis. Note that this usage of the term ''linear'' is not the same as in the section above, because linear polynomials over the real numbers do not in general satisfy either additivity or homogeneity. In fact, they do so
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
. Hence, if , the function is often called an affine function (see in greater generality
affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, ...
).


Boolean functions

In
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
, a linear function is a function f for which there exist a_0, a_1, \ldots, a_n \in \ such that :f(b_1, \ldots, b_n) = a_0 \oplus (a_1 \land b_1) \oplus \cdots \oplus (a_n \land b_n), where b_1, \ldots, b_n \in \. Note that if a_0 = 1, the above function is considered affine in linear algebra (i.e. not linear). A Boolean function is linear if one of the following holds for the function's
truth table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional argumen ...
: # In every row in which the truth value of the function is T, there are an odd number of Ts assigned to the arguments, and in every row in which the function is F there is an even number of Ts assigned to arguments. Specifically, , and these functions correspond to
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
s over the Boolean vector space. # In every row in which the value of the function is T, there is an even number of Ts assigned to the arguments of the function; and in every row in which the
truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (''true'' or '' false''). Computing In some progr ...
of the function is F, there are an odd number of Ts assigned to arguments. In this case, . Another way to express this is that each variable always makes a difference in the
truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (''true'' or '' false''). Computing In some progr ...
of the operation or it never makes a difference.
Negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
, Logical biconditional, exclusive or, tautology, and contradiction are linear functions.


Physics

In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, ''linearity'' is a property of the
differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
governing many systems; for instance, the
Maxwell equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. Th ...
or the
diffusion equation The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's la ...
. Linearity of a homogenous
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
means that if two functions ''f'' and ''g'' are solutions of the equation, then any linear combination is, too. In instrumentation, linearity means that a given change in an input variable gives the same change in the output of the measurement apparatus: this is highly desirable in scientific work. In general, instruments are close to linear over a certain range, and most useful within that range. In contrast, human senses are highly nonlinear: for instance, the brain completely ignores incoming light unless it exceeds a certain absolute threshold number of photons.


Electronics

In
electronics The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...
, the linear operating region of a device, for example a
transistor upright=1.4, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink). A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch e ...
, is where an output
dependent variable Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand ...
(such as the transistor collector current) is directly proportional to an input dependent variable (such as the base current). This ensures that an analog output is an accurate representation of an input, typically with higher amplitude (amplified). A typical example of linear equipment is a high fidelity
audio amplifier An audio power amplifier (or power amp) is an electronic amplifier that amplifies low-power electronic audio signals, such as the signal from a radio receiver or an electric guitar pickup, to a level that is high enough for driving loudspea ...
, which must amplify a signal without changing its waveform. Others are
linear filter Linear filters process time-varying input signals to produce output signals, subject to the constraint of linearity. In most cases these linear filters are also time invariant (or shift invariant) in which case they can be analyzed exactly using ...
s, and
linear amplifier A linear amplifier is an electronic circuit whose output is proportional to its input, but capable of delivering more power into a load. The term usually refers to a type of radio-frequency (RF) power amplifier, some of which have output power mea ...
s in general. In most scientific and technological, as distinct from mathematical, applications, something may be described as linear if the characteristic is approximately but not exactly a straight line; and linearity may be valid only within a certain operating region—for example, a high-fidelity amplifier may distort a small signal, but sufficiently little to be acceptable (acceptable but imperfect linearity); and may distort very badly if the input exceeds a certain value.


Integral linearity

For an electronic device (or other physical device) that converts a quantity to another quantity, Bertram S. Kolts writes:
There are three basic definitions for integral linearity in common use: independent linearity, zero-based linearity, and terminal, or end-point, linearity. In each case, linearity defines how well the device's actual performance across a specified operating range approximates a straight line. Linearity is usually measured in terms of a deviation, or non-linearity, from an ideal straight line and it is typically expressed in terms of percent of full scale, or in ppm (parts per million) of full scale. Typically, the straight line is obtained by performing a least-squares fit of the data. The three definitions vary in the manner in which the straight line is positioned relative to the actual device's performance. Also, all three of these definitions ignore any gain, or offset errors that may be present in the actual device's performance characteristics.


Military tactical formations

In military tactical formations, "linear formations" were adapted starting from phalanx-like formations of
pike Pike, Pikes or The Pike may refer to: Fish * Blue pike or blue walleye, an extinct color morph of the yellow walleye ''Sander vitreus'' * Ctenoluciidae, the "pike characins", some species of which are commonly known as pikes * ''Esox'', genus of ...
protected by handgunners, towards shallow formations of handgunners protected by progressively fewer pikes. This kind of formation got progressively thinner until its extreme in the age of Wellington's ' Thin Red Line'. It was eventually replaced by skirmish order when the invention of the
breech-loading A breechloader is a firearm in which the user loads the ammunition (cartridge or shell) via the rear (breech) end of its barrel, as opposed to a muzzleloader, which loads ammunition via the front ( muzzle). Modern firearms are generally breech ...
rifle A rifle is a long-barreled firearm designed for accurate shooting, with a barrel that has a helical pattern of grooves ( rifling) cut into the bore wall. In keeping with their focus on accuracy, rifles are typically designed to be held with ...
allowed soldiers to move and fire in small, mobile units, unsupported by large-scale formations of any shape.


Art

Linear is one of the five categories proposed by Swiss art historian Heinrich Wölfflin to distinguish "Classic", or
Renaissance art Renaissance art (1350 – 1620 AD) is the painting, sculpture, and decorative arts of the period of European history known as the Renaissance, which emerged as a distinct style in Italy in about AD 1400, in parallel with developments which occ ...
, from the
Baroque The Baroque (, ; ) is a style of architecture, music, dance, painting, sculpture, poetry, and other arts that flourished in Europe from the early 17th century until the 1750s. In the territories of the Spanish and Portuguese empires including t ...
. According to Wölfflin, painters of the fifteenth and early sixteenth centuries (
Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially res ...
,
Raphael Raffaello Sanzio da Urbino, better known as Raphael (; or ; March 28 or April 6, 1483April 6, 1520), was an Italian painter and architect of the High Renaissance. List of works by Raphael, His work is admired for its clarity of form, ease of ...
or
Albrecht Dürer Albrecht Dürer (; ; hu, Ajtósi Adalbert; 21 May 1471 – 6 April 1528),Müller, Peter O. (1993) ''Substantiv-Derivation in Den Schriften Albrecht Dürers'', Walter de Gruyter. . sometimes spelled in English as Durer (without an umlaut) or Due ...
) are more linear than "
painterly Painterliness is a concept based on ''german: malerisch'' ('painterly'), a word popularized by Swiss art historian Heinrich Wölfflin (1864–1945) to help focus, enrich and standardize the terms being used by art historians of his time to cha ...
" Baroque painters of the seventeenth century (
Peter Paul Rubens Sir Peter Paul Rubens (; ; 28 June 1577 – 30 May 1640) was a Flemish artist and diplomat from the Duchy of Brabant in the Southern Netherlands (modern-day Belgium). He is considered the most influential artist of the Flemish Baroque traditio ...
,
Rembrandt Rembrandt Harmenszoon van Rijn (, ; 15 July 1606 – 4 October 1669), usually simply known as Rembrandt, was a Dutch Golden Age painter, printmaker and draughtsman. An innovative and prolific master in three media, he is generally consid ...
, and Velázquez) because they primarily use outline to create
shape A shape or figure is a graphics, graphical representation of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type. A pl ...
. Linearity in art can also be referenced in
digital art Digital art refers to any artistic work or practice that uses digital technology as part of the creative or presentation process, or more specifically computational art that uses and engages with digital media. Since the 1960s, various names ...
. For example,
hypertext fiction Hypertext fiction is a genre of electronic literature, characterized by the use of hypertext links that provide a new context for non-linearity in literature and reader interaction. The reader typically chooses links to move from one node of text t ...
can be an example of
nonlinear narrative Nonlinear narrative, disjointed narrative, or disrupted narrative is a narrative technique, sometimes used in literature, film, video games, and other narratives, where events are portrayed, for example, out of chronological order or in other ways ...
, but there are also websites designed to go in a specified, organized manner, following a linear path.


Music

In music the linear aspect is succession, either intervals or
melody A melody (from Greek language, Greek μελῳδία, ''melōidía'', "singing, chanting"), also tune, voice or line, is a Linearity#Music, linear succession of musical tones that the listener perceives as a single entity. In its most liter ...
, as opposed to
simultaneity Simultaneity may refer to: * Relativity of simultaneity, a concept in special relativity. * Simultaneity (music), more than one complete musical texture occurring at the same time, rather than in succession * Simultaneity, a concept in Endogeneit ...
or the vertical aspect.


In statistics


See also

* Linear actuator *
Linear element Electrical elements are conceptual abstractions representing idealized electrical components, such as resistors, capacitors, and inductors, used in the analysis of electrical networks. All electrical networks can be analyzed as multiple electrica ...
*
Linear foot The foot ( feet), standard symbol: ft, is a unit of length in the British imperial and United States customary systems of measurement. The prime symbol, , is a customarily used alternative symbol. Since the International Yard and ...
*
Linear system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstraction o ...
*
Linear programming Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear function#As a polynomial function, li ...
* Linear differential equation * Bilinear * Multilinear *
Linear motor A linear motor is an electric motor that has had its stator and rotor "unrolled", thus, instead of producing a torque (rotation), it produces a linear force along its length. However, linear motors are not necessarily straight. Characteristicall ...
* Linear A and
Linear B Linear B was a syllabic script used for writing in Mycenaean Greek, the earliest attested form of Greek. The script predates the Greek alphabet by several centuries. The oldest Mycenaean writing dates to about 1400 BC. It is descended from ...
scripts. * Linear interpolation


References


External links

*{{wiktionary-inline Elementary algebra Physical phenomena Broad-concept articles